3.295 \(\int F^{a+b (c+d x)^3} \, dx\)

Optimal. Leaf size=47 \[ -\frac{F^a (c+d x) \text{Gamma}\left (\frac{1}{3},-b \log (F) (c+d x)^3\right )}{3 d \sqrt [3]{-b \log (F) (c+d x)^3}} \]

[Out]

-(F^a*(c + d*x)*Gamma[1/3, -(b*(c + d*x)^3*Log[F])])/(3*d*(-(b*(c + d*x)^3*Log[F]))^(1/3))

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Rubi [A]  time = 0.0070726, antiderivative size = 47, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {2208} \[ -\frac{F^a (c+d x) \text{Gamma}\left (\frac{1}{3},-b \log (F) (c+d x)^3\right )}{3 d \sqrt [3]{-b \log (F) (c+d x)^3}} \]

Antiderivative was successfully verified.

[In]

Int[F^(a + b*(c + d*x)^3),x]

[Out]

-(F^a*(c + d*x)*Gamma[1/3, -(b*(c + d*x)^3*Log[F])])/(3*d*(-(b*(c + d*x)^3*Log[F]))^(1/3))

Rule 2208

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_)), x_Symbol] :> -Simp[(F^a*(c + d*x)*Gamma[1/n, -(b*(c + d*x)
^n*Log[F])])/(d*n*(-(b*(c + d*x)^n*Log[F]))^(1/n)), x] /; FreeQ[{F, a, b, c, d, n}, x] &&  !IntegerQ[2/n]

Rubi steps

\begin{align*} \int F^{a+b (c+d x)^3} \, dx &=-\frac{F^a (c+d x) \Gamma \left (\frac{1}{3},-b (c+d x)^3 \log (F)\right )}{3 d \sqrt [3]{-b (c+d x)^3 \log (F)}}\\ \end{align*}

Mathematica [A]  time = 0.0160868, size = 47, normalized size = 1. \[ -\frac{F^a (c+d x) \text{Gamma}\left (\frac{1}{3},-b \log (F) (c+d x)^3\right )}{3 d \sqrt [3]{-b \log (F) (c+d x)^3}} \]

Antiderivative was successfully verified.

[In]

Integrate[F^(a + b*(c + d*x)^3),x]

[Out]

-(F^a*(c + d*x)*Gamma[1/3, -(b*(c + d*x)^3*Log[F])])/(3*d*(-(b*(c + d*x)^3*Log[F]))^(1/3))

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Maple [F]  time = 0.017, size = 0, normalized size = 0. \begin{align*} \int{F}^{a+b \left ( dx+c \right ) ^{3}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(F^(a+b*(d*x+c)^3),x)

[Out]

int(F^(a+b*(d*x+c)^3),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int F^{{\left (d x + c\right )}^{3} b + a}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(F^(a+b*(d*x+c)^3),x, algorithm="maxima")

[Out]

integrate(F^((d*x + c)^3*b + a), x)

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Fricas [A]  time = 1.5441, size = 157, normalized size = 3.34 \begin{align*} \frac{\left (-b d^{3} \log \left (F\right )\right )^{\frac{2}{3}} F^{a} \Gamma \left (\frac{1}{3}, -{\left (b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3}\right )} \log \left (F\right )\right )}{3 \, b d^{3} \log \left (F\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(F^(a+b*(d*x+c)^3),x, algorithm="fricas")

[Out]

1/3*(-b*d^3*log(F))^(2/3)*F^a*gamma(1/3, -(b*d^3*x^3 + 3*b*c*d^2*x^2 + 3*b*c^2*d*x + b*c^3)*log(F))/(b*d^3*log
(F))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int F^{a + b \left (c + d x\right )^{3}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(F**(a+b*(d*x+c)**3),x)

[Out]

Integral(F**(a + b*(c + d*x)**3), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int F^{{\left (d x + c\right )}^{3} b + a}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(F^(a+b*(d*x+c)^3),x, algorithm="giac")

[Out]

integrate(F^((d*x + c)^3*b + a), x)